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Faithful representations of Banach-Lie algebras. (English) Zbl 0597.46050
It is shown here that given a Banach-Lie algebra (A,[, ], there is a Banach space E and a continuous linear injection j of A into the algebra L(E) of all bounded linear operators on E such that $$j[x,y]=j(x)j(y)- j(y)j(x)=[j(x),j(y)]$$, for all x,y$$\in A$$. This is a weaker version of Ado’s result for the finite-dimensional Banach -Lie algebra case [for example, see B. Maissen, Acta Math. 108, 229-270 (1962; Zbl 0207.337)].
Reviewer: T.Husain
##### MSC:
 46H15 Representations of topological algebras 46H10 Ideals and subalgebras 17B65 Infinite-dimensional Lie (super)algebras
##### Keywords:
Banach-Lie algebra; continuous linear injection